Isometries Preserve Distances

Table of Contents

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1. Synthetic and Analytic Geometry

In his Erlangen Program, the great 19th century German
Geometer, Felix Klein, proposed that the basis of geometry
should be groups of point transformations. Euclid’s geometry
was based on axioms that made assertions about primitive
geometrical concepts, like point, line, angle etc which
were considered to need no definition since "everybody" agreed
on what they were anyway. All subsequent geometrical knowledge
had to be derived logically from the axioms. We call this an
axiomatic system.

Euclid’s program had flaws that were not
completely resolved until the end of the 19th century.
Applied to geometry, the axiomatic method developed into
what is called synthetic geometry,
where the "undefined terms" became abstractions, the axioms
became sentences relating the undefined terms. As is taught
in any course on the subject (such as MA 402 at Illinois),
things become interesting only if the undefined terms
and their relations are interpreted in a familiar setting,
in which the axioms can be checked to be true or not.

In contrast to synthetic geometry, analytic geometry, which
derives from the work of Descartes and Fermat, is based on
the properties of numbers. You can, if you wish, build up the
number system axiomatically (as is done in pute mathematics) or you
can just accept numbers as you learned about them in school.
Now, geometric objects are sets of numbers, and their relations
are expressed in terms of set theory. Of course we still think
geometrically about them, but consistency is derived ultimately
from the number systems and their algebra. This is the geometry
you learned about first in high school, and later added the
knowledge of vectors and their properties in calculus.

The dispute as to which approach is "better", the synthetic or
the analytic, was resolved by David Hilbert (1900), who proved through
logic, that each can be derived from the other. Thus, if the
number systems have logical errors, then these will show up in
the geometry too, and vice versa. Few mathematicians worry about
these issues nowadays, since there are more important things to
discover and deeper controversies to resolve.

One of Euclid’s primitives was the idea of congruence. When are
two figure the "same" in a geometrical sense. You should recall
such criteria as side-angle-side (SAS), which were theorems for
Euclid, but his "proof" was not correct. In synthetic geometry,
congruence remains undefined, and SAS becomes an axiom.

In analytic geometry, as we shall see below, a congruence is a
point transformation (points go to points) which preserves
length. More precisely, the distance between any two points
remains the same after as it was before the transformation. We
define distance in terms of the familiar formula of the
Pythagorean theorem, which we express in coordinate and in vector
form, where that $ X = (x,y), W =(u,v) $

2. Background

Recall that
translations and dilatations
are point transformations of the Euclidean plane with
several important properties.
In particular, displacement vectors are preserved (by
translations or scaled dilatations. That is, if
`tau = tau_A` and `delta=delta_{Q,r}`,
and we write the image of a point under a transformation by
decorating its name with a superscript, then

`X^tau -W^tau =X-W`

`X^delta - W^delta =r(X-W)`.

Question 2.

Do you recall the formulas for a translation and a dilatation? If
you do, or have to look them up, then prove this assertion on your
scratch pad by the side of your computer!

From the way these transformations affect displacements we see that
translations always preserve distance. So these are definitely isometries.
For dilatations $ r = \pm 1 $ will yield isometries. The "plus"
case is uninteresting, in the sense that the answer is obvious.
Recall that the more interesting "minus" case is
for the central reflection, discussed earlier.

2.1. Length, Norm, and Magnitude

Geometrically, these are all the same non-negative real number
associated with a vector. We like to use length when we represent
the vector as one of a collection of mutually parallel "arrows" in
the plane. The norm is a more abstract concept in higher algebra.
We tend to use magnitude in the same breath as direction. Any
physical object that has magnitude and direction can be represented
as a vector. For example, velocity and acceleration have magnitude
and direction, and therefore are vector quantities in physics.

The magnitude is intimately connected with the dot product, and
in a sense they are equivalent concepts.

Comment on Notation

It is generally inconvenient to use any special symbol for the
dot product, unless other products involving vectors are used in
the same discussion. It detracts from the ability to scan algebraic
manipulations you worked so hard in high school to master.

So we shall, when no ambiguity threatens, just
write $ XY $ instead of $ X $ • $ Y $. This allows us
to use powers, as in $ X X = X^2 $.

The commutative and distributive properties of numbers carries
over to vectors. Just remember that $ XY $ is a number,
not a vector. Here is a list of properties you learned in calculus.

For vectors $ X,Y,Z, O $ and scalars $ r, s, 1, 0 $
we have the

2.2. Bilinearity Properties of the Dot Product

Formula

Property name

$ X Y = Y X $

commutative for dot product

$ X(Y+Z)= XY + XZ $

distributive for dot product

$ (rX)Y = r(XY) $

associative for scalar product

$ (r+s)X = rX + sX $

distributive for scalar product

$ 1X = X $

scalar product with 1

$ 0X = O $

scalar product with 0

Note the associative property does not hold: $ (XY)Z \ne X(YZ) $.
The LHS is a vector parallel to $ Z $, and the RHS has the
same direction as $ X $. Also note that when it is important
to distinguish between the scalar zero $ 0 $ and the
zero vector, $ O $, we follow the Tondeur’s convention of using
the same letter for the point int he plane, and the vector defined by
the displacement from the origin to that point. When these distinctions
are not obvious from the context, and it is important to emphasize them,
you can underline vectors, or stick a little arrow on top of them.
The less notational fuss the better, especially in emails.

Definition.

The magnitude `|A|` of a vector is given by
`|A|^2=A A = A^2`.

For a displacement vector `B-A`, we have
`|B-A|^2=` `(B-A)(B-A)`.
By the bilinearity of the dot product, we can write this as
`(B-A)^2=B^2-2AB+A^2`. From this we see we could
"define" the dot product itself in terms of norms.

The length of a displacement vector between two points has the
following properties:

L1: `|B-A| >= 0`.

L2: `|B-A|=0` if and only if `B=A`.

L3: `|C-A| <= |C-B|+|B-A|`.

Note that these are precisely the properties of the distance between
two points. Property L3 is called the triangle inequality because
any side of a triangle is no longer than the sum of the other two
sides.

The first would say that $ \alpha $ preserves the entire displacement
vector, body and soul, namely the direction as well as the magnitude.
The second confuses the displacement of the images of two points, with the
image of the displacement of two points. The displacement $ X-Y $
is a vector. A transformation is defined on points. So $ (X-Y)^\alpha $
can only mean the image of the point $ (X-Y) $.

A counterexample for the second goes as follows.
Let $ X^\alpha = X +D $ be a translation which is not the
identity. Then $ \| (X-Y)^\alpha\| = \| X-Y+D\| $.
Now take the special case that $ X = Y $ and conclude that
$\| D\| = 0 $, contrary to our assumption.

Translations preserve displacement vectors, and central reflections reverse
displacement vectors. Thus both of these point transformations are isometries.

Since `B^delta - A^delta =r(B-A) rArr |B^delta - A^delta|=|r(B-A)|=|r|\ |B-A|`, `delta` is an isometry if and only if `r=+-1`.

Another example of an isometry is a rotation `rho_{Q,theta}`
about the point `Q` by `theta` degrees. This is intuitively
obvious, since rotations are rigid motions, like translations. We will study
rotations more thoroughly in a subsequent lesson.

We end this section with a proposition that has an easy proof.

Proposition.

The composition of two isometries is again an isometry.

Proof. Suppose `alpha` and
`beta` are two isometries and let `gamma=beta alpha` be their composition.
Then for any pair of points,

`|X^gamma - Y^gamma|\ ` =

`|beta alpha(X) - beta alpha(Y)|`

=

`|beta(X^alpha) - beta(Y^alpha)|`

=

`|X^alpha - Y^alpha|`

(`beta` is an
isometry )

=

`|X-Y|`

(`alpha` is an isometry).

Comment.

Note that we pay a price for using the convenient superscript notation
for point transformations. Compositions in functional notations have
to be read from right to left, while in the exponent they read from
left to right: $ \alpha \beta (X) = (X^\beta)^\alpha $.

Moreover, above we defined isometries to be point transformations,
and hence we assume they are 1:1 and onto although we will prove that
this is a consequence of the distance preserving property. We could,
with Tondeur, just assume that isometries are transformations of the
plane that takes points to points, and derive the bijectivity.

Recall that for subsets of the point transformations to be a groups,
in addition to closure which we have just proved, we
only need to check that inverses preserve distance.

where $ =_{bb{1}} $ is true because $ \alpha \omega = \iota $, and
$ =_{bb{2}} $ follows because $ \alpha $ is given to be an isometry.

We conclude this lesson by observing that we have proved this theorem.

Theorem.

The set of all isometries is a group under composition.

Comment.

In Tondeur’s text the proof of this theorem is considerably more complicated
than we have given here. This is true only because we make the initial
stipulation that an isometry is already a point-transformation, and that such
transformations are 1:1 and onto. See Tondeur’s argument that it is, in
fact, not necessary to make this assumption. But it shortens the exposition.